×

The Dehn function of Richard Thompson’s group \(F\) is quadratic. (English) Zbl 1166.20024

Summary: We prove that the Dehn function (that is, the smallest isoperimetric function) of R. Thompson’s group \(F\) is quadratic.

MSC:

20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory
57M07 Topological methods in group theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baumslag, G., Miller, C.F., Short, H.: Isoperimetric inequalities and the homology of groups. Invent. Math. 113, 531–560 (1993) · Zbl 0829.20053
[2] Brin, M.G., Squier, C.C.: Groups of piecewise linear homeomorphisms of the real line. Invent. Math. 79, 485–498 (1985) · Zbl 0563.57022
[3] Brown, K.S.: Finiteness properties of groups. J. Pure Appl. Algebra 44, 45–75 (1987) · Zbl 0613.20033
[4] Brown, K.S., Geoghegan, R.: An infinite-dimensional torsion-free FPgroup. Invent. Math. 77, 367–381 (1984) · Zbl 0557.55009
[5] Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word processing in groups. Boston, London: Jones and Barlett 1992 · Zbl 0764.20017
[6] Cannon, J.W., Floyd, W.J., Parry, W.R.: Introductory notes on Richard Thompson’s groups. Enseign. Math., II Sér. 42, 215–256 (1996) · Zbl 0880.20027
[7] Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: Handbook of Theoretical Computer Science, ed. by J. van Leeuwen, Chap. 6, pp. 244–320. Elsevier Science Publishers B.V. 1990
[8] Gersten, S.: Isoperimetric and isodiametric functions of finite presentations. In: Geometric group theory, Vol. 1 (Sussex 1991). Lond. Math. Soc. Lect. Note Ser. 181, 79–96 (1993) · Zbl 0829.20054
[9] Gersten, S.: Thompson’s group F is not combable. Unpublished
[10] Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, ed. by S. Gersten, MSRI Publ., Vol. 8, pp.75–263. Springer 1987
[11] Guba, V.S.: Polynomial upper bounds for the Dehn function of R. Thompson’s group F. J. Group Theory 1, 203–211 (1998) · Zbl 0907.20037
[12] Guba, V.S.: Polynomial Isoperimetric Inequalities for Richard Thompson’s Groups F, T, and V. In: Algorithmic Problems in Groups and Semigroups, ed. by J.-C. Birget et al., pp. 91–120. Boston, Basel, Berlin: Birkhäuser 2000
[13] Guba, V.S., Sapir, M.V.: Diagram groups. Mem. Am. Math. Soc. 130, 1–117 (1997) · Zbl 0930.20033
[14] Guba, V.S., Sapir, M.V.: The Dehn function and a regular set of normal forms for R. Thompson’s group F. J. Aust. Math. Soc., Ser. A 62, 315–328 (1997) · Zbl 0899.20013
[15] Lyndon, R., Schupp, P.: Combinatorial group theory. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0368.20023
[16] Madlener, K., Otto, F.: Pseudo-natural algorithms for the word problem for finitely presented monoids and groups. J. Symb. Comput. 1, 383–418 (1985) · Zbl 0591.20038
[17] Papasoglu, P.: On the asymptotic cone of groups satisfying a quadratic isoperimetric inequality. J. Differ. Geom. 44, 789–806 (1996) · Zbl 0893.20029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.